# Nonlinear waves at interfaces contributions

*To be updated until april 30th*

## scikit-fdiff, a new tool for PDE solving

Scikit-FDiff (formerly known as Triflow) is a new tool, written in pure Python, that focus on reducing the time between the developpement of the mathematical model and the numerical solving. It allows an easy and automatic finite difference discretization, thanks to a symbolic processing that can deal with systems of multi-dimensional partial differential equation with complex boundary conditions.

Using finite differences and the method of lines, it allows the transformation of the original PDE into an ODE, providing a fast computation of the temporal evolution vector and the Jacobian matrix. The later is pre-computed in a symbolic way and sparse by nature. It can be evaluated with as few computational resources as possible, allowing the use of implicit and explicit solvers at a reasonable cost.

Classic ODE solvers have been implemented (or made available from dedicated python libraries), including backward and forward Euler scheme, Crank-Nickolson, explicit Runge-Kutta. More complexes ones, like improved Rosenbrock-Wanner schemes up to the 6th order, are also available. The time-step is managed by a built-in error computation, which ensures the accuracy of the solution. The main goal of the software is to minimize the time spent writting numerical solvers to focus on model development and data analysis.

Scikit-Fdiff is then able to solve toy cases in a few line of code as well as complex models. Extra tools are available, such as data saving during the simulation, real-time plotting and post-processing. It has been validated with the shallow-water equation on dam-breaks and the steady-lake case. It has also been applied to heated falling-films, dropplet spread and simple moisture flow in porous medium.

## Nonlinear, short-crested and localized waves

## On an improved model for long internal gravity waves

## Experimental evidence of hydrodynamic instantons and their unifying role in the theory of rogue waves

## Interfacial nonlinearities to damp sloshing waves

## Birth of a hydraulic jump

Alexis Duchesne, Tomas Bohr and Anders Andersen

The hydraulic jump, i.e., the sharp transition between a supercritical and a subcritical free-surface flow, has been extensively studied. However, an important question has been left unanswered: How does a hydraulic jump form? We present here an experimental and theoretical study of the formation of stationary hydraulic jumps in centimeter-sized channels.

We start with an empty channel and then change the flow rate abruptly from zero to a constant value. This leads to the formation of a stationary hydraulic jump in a two stage process. Firstly, the channel fills quickly ($\sim 1$ s). Initially the liquid layer shows a linearly increasing height profile and a front position with a square root dependence on time. When the height of the liquid front reaches a critical value, it remains constant throughout the rest of the filling process. At low flow rate the jump forms during the filling of the channel whereas the jump appears at a later stage when the flow rate is high. Secondly, the influence of the downstream boundary condition makes the jump move slowly ($\sim 10$ s) upstream to its final position with exponentially decreasing speed.

## Hypergravity Wave Turbulence

## Internal gravity waves generated by turbulent flows

Many geophysical and astrophysical fluids, including planetary atmospheres, stars and oceans, consist of turbulent flows adjacent to stably-stratified fluid layers. Because waves can drive large-scale flows, increase scalar mixing and are sometimes easier to observe than turbulent motions, two important questions for these fluids are: how much energy goes from the turbulence into internal waves in the stable layer? What kind of waves (i.e. what wavenumbers and frequencies) are generated most efficiently?

In this talk we will answer these two questions by presenting a theoretical prediction for the energy flux spectrum of waves generated by turbulent convection and comparing it with results from 3D direct numerical simulations (DNS) of self-organised convective--stably-stratified fluids. We will show that DNS and theory agree well for the range of strong turbulence-strong stratification parameters tested, giving some confidence in the analytical expression for the energy flux spectrum of the waves, which is based on a theory that assumes waves are generated by Reynolds stresses due to eddies in the turbulent region. We hope that our results will help quantify wave generation in geophysical and astrophysical fluids.

## Leidenfrost Effect: The life of a levitating water droplet on a hot vapour layer

The Leidenfrost effect is a physical phenomenon in which a liquid droplet floats on its own evaporating vapour due to the presence of a hot substrate underneath. This effect was discovered by Johan Leidenfrost in 1771 and investigated by John Tyndall as narrated in his book “Heat: a mode of motion (1875). Leidenfrost droplets constitutes an interesting out of equilibrium system which can be a nice playground for laboratory experiments on capillarity and fluid motion. In my talk, I will review the recent experimental and theoretical studies that we have undergone in our laboratory. I will discuss the behaviour of Leidenfrost droplets in the super-levitation regime [1,2] which takes place for a very small droplet radius and reveals the signature of the end of the lubrication regime. I will also discuss a new technique for generating Leidenfrost droplet at ambient temperature (20 Celcius) by using a low-pressure atmosphere [3]. These droplets could have applications as micro-reactors. Finally, I will expose theoretical and experimental results on Leidenfrost droplets which are confined in a two-dimensional geometry by means of a Hele-Shaw cell [4,5], in particular their oscillations and the dynamics of a growing hole. Finally, I will conclude with some questions on their interface fluctuations when the system is close to the Leidenfrost transition.

[1] Take off of small Leidenfrost droplets, F Celestini, T Frisch, Y Pomeau, Physical review letters 109 (2012)

[2] The Leidenfrost effect: From quasi-spherical droplets to puddles, Y Pomeau, M Le Berre, F Celestini, T Frisch, Comptes Rendus Mecanique 340 (2012)

[3] Room temperature water Leidenfrost droplets, F Celestini, T Frisch, Y Pomeau Soft Matter 9 (2013)

[4]Two-dimensional Leidenfrost droplets in a Hele-Shaw cell, F Celestini, T Frisch, A Cohen, C Raufaste, L Duchemin, Y Pomeau, Physics of Fluids 26 (2014)

[5] Hole growth dynamics in a two-dimensional Leidenfrost droplet, C Raufaste, F Celestini, A Barzyk, T Frisch, Physics of Fluids 27 (2015)

## Singularity turbulence

## Energy cascade in internal wave attractors

## Resonances of Internal Gravity Waves in Stratified Shear Flows

P. Le Gal, G. Facchini, J. Chen, S. Le Dizès, M. Le Bars, B. Favier, IRPHE, Aix Marseille Univ., CNRS, Centrale Marseille, France

U. Harlander, I.D. Borcia, Dept. of Aerodynamics and Fluid Mechanics Brandenburg Univ. of Technology, Cottbus, Germany

W. Meng, Dept. of Mechanical Engineering, Univ. of California, Berkeley, CA 94709, USA

We will present here a new instability mechanism that affects the Plane Couette flow and the Plane Poiseuille flow when these flows are stably stratified in density along the vertical direction, i.e. orthogonal to the horizontal shear. Stratified shear flows are ubiquitous in nature and in a geophysical context, we may think to water flows in submarine canyons, to winds in deep valleys, to currents along sea shores or to laminar flows in canals where density stratification can be due to temperature or salinity gradients. Our study is based on two sets of laboratory experiments with salt stratified water flows, on linear stability analyses and on direct numerical simulations. It follows recent investigations of instabilities in stratified rotating or non rotating shear flows: the stratorotational instability [2],[3], the stratified boundary layer instability [4] where it was shown that these instabilities belong to a class of instabilities caused by the resonant interaction of Doppler shifted internal gravity waves. Our laboratory experiments for Plane Couette and Plane Poiseuille flows, based on visualizations and PIV measurements, show in both cases the appearance of braided wave patterns when the experimental parameters, depending on the Reynolds and Froude numbers, are above a threshold. The non linear saturation of the instability leads to a meandering in the horizontal plane arranged in layers stacked along the vertical direction [5]. Comparison with theoretical predictions for the instability threshold and the critical wavenumbers calculated by linear analysis is excellent. Moreover, direct numerical simulations permit to complete the description of this instability that can be interpreted as a resonant interaction of boundary trapped waves [6].

[1] S. Orszag, JFM 50(4), 689-703, 1971.

[2] M. Le Bars & P. Le Gal, Phys. Rev. Lett. 99, 064502, 2007.

[3] G. Rüdiger, T. Seelig, M. Schultz, M. Gellert, Ch. Egbers & U. Harlander, GAFD,111, 429-447, 2017.

[4] J. Chen, Y. Bai, & S. Le Dizès, JFM 795, 262-277, 2016.

[5] D. Lucas, C.P. Caulfield, R. R. Kerswell, arXiv:1808.01178, 2019.

[6] G. Facchini, B. Favier, P. Le Gal, M. Wang, M. Le Bars, JFM 853, 205-234, 2018.

## Transition from inertial wave turbulence to geostrophic turbulence in rotating fluids - an experimental study

We present an experimental investigation of the turbulent saturation of the flow driven by parametric resonance of inertial waves in a rotating fluid. In our setup, a half-meter wide ellipsoid filled with water is brought to solid body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is supplied through the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow main feature is a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly non-linear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al, PRL 2017 who introduced the possibility of these two turbulent regimes. The transition between these two regimes and there relevance to geophysical applications are finally discussed.

## Circular hydraulic jump and inclined jump

We have investigated the flow and interface structures involved in a circular hydraulic jump formed by impacting a large horizontal disk with a jet of viscous liquid. Among other results, we found that the Froude number at the jump entry seems to be locked to a critical, constant value. This empirical condition, when combined with the large scale lubrication flow structure leads to a “à la Bohr” scaling, with Logarithmic corrections that can be explicitly calculated, in agreement with recent theoretical and numerical modeling. In a second step, we have investigated the jump structure formed when the jet and the impacted disk are inclined of the same amount, after varying the wetting conditions on the disk (hydrophilic, partial wetting and superhydrophobic). The results are very sensitive to the wetting properties as well as to the flow rate and plate inclination. We have tried to interpret the scaling laws observed with simple models generalizing Watson approach of the circular hydraulic jump.

## Statistics and models for Faraday pilot waves

## Wave turbulence at the surface of water: the role of bound waves on intermittency

## Bacteria display optimal transport near surfaces

## Intermittency and Leray singularities

## Spin lattices of walking droplets

A droplet bouncing on the surface of a vibrating liquid bath can self-propel across the surface through interaction with the wave field it generates by bouncing. These walking droplets or “walkers” comprise a droplet and its guiding wave, and have been shown to exhibit several behaviors analog to quantum systems. Most analogs consider a single walker interacting with boundaries or experiencing external forces. Controlling multiple walkers is challenging as their continuous wave-mediated interactions usually lead to pair bound states and droplet-droplet coalescence. Here I show that multiple walkers can be manipulated by designing the bottom topography of the vibrating bath as a lattice composed of deeper regions separated by shallow regions. Specifically, I show that circular wells at the bottom of the fluid bath encourage individual droplets to walk in clockwise or counter-clockwise direction along circular trajectories centered at the lattice sites. A thin fluid layer between the wells enables wave-mediated interactions between neighboring walkers resulting in ordered rotation dynamics across the lattice. When the pair coupling is sufficiently strong, interactions between neighboring droplets may induce local spin flips leading to ferromagnetic or anti-ferromagnetic order. In addition, an anti-ferromagnetic to ferromagnetic transition is obtained when the whole bath is rotating. Our experiments demonstrate the spontaneous emergence of collective behavior of walkers that mimic spin lattices.

This work has been done at Massachusetts Institute of Technology with Pedro J. Saenz, Sam E. Turton, Alexis Goujon, Rodolfo R. Rosales, Jörn Dunkel and John W. M. Bush.

## When wind waves become Francis solitons

## Nonlinear waves in Plateau borders

Plateau borders are the liquid microchannels found at the intersection between bubbles inside liquid foams. They concentrate most of the mass and their role is essential to account for the foams drainage and mechanical properties. During this presentation, experiments and results will be shown about the relaxation of a single Plateau border that is subject to external perturbations. We will see how a negative effective surface tension drives the dynamics, with a special emphasis on regimes dominated by inertial flows and nonlinear waves.

## Sheared falling film flows: a numerical study

## A simplified model of aquatic locomotion

We have developed a simple model of aquatic locomotion. Using the theory of complex variables, we have estimated the hydrodynamic forces acting on an infinite thin rigid plate of length L, following the seminal Work of Theordorsen [1].

By considering the different possible motions of the swimmer, we calculate the velocity potential to derive the pressure by means of the generalised Bernoulli relation. We show that the effect of flow unsteadiness is the principal mechanism for locomotion [2].

We impose a periodic rotation of the tail in order to approximate the undulatory motion of the swimmer. We show the linear dependence of longitudinal velocity on the angular frequency predicted by Gazzola et al [3] . We also predict that the transverse motion presents the same frequency as the forcing whereas the longitudinal motion is a linear function of time plus a periodic term with double frequency.

Finally, by taking the angle of the tail as a small parameter we perform a perturbative expansion to obtain an equation linking swimming velocity to the different parameters involved in swimming. The results arised from this perturbative method are in high accordance with the numerical results.

[1] Theodorsen, T., General theory of aerodynamic instability and the mechanism of flutter, NACA TR No. 496, 1934

[2] Garrick, I. E., Propulsion of a flapping and oscillating airfoil, NACA TR No. 567, 1936

[3] Gazzola, M., Argentina, M., & Mahadevan, L , Scaling macroscopic aquatic locomotion, Nature Physics 10 (10), 758-761, 2014